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Long-term and short-term asymptotes of reaction rate constants in the presence of reactivity anisotropy

Abstract

Based on the general theory of bimolecular multistage diffusion-influenced chemical reactions proceeding from different sites in solutions available in literature some general model of reactivity anisotropy is formulated. It treats active patches on the surfaces of rigid spheres as sites and, in the general case, includes into consideration relative translational motion of reactants and their rotational relaxation. For this model, general expressions are obtained for averaged free resolvents necessary for calculation of rate constants of multistage multisite diffusion-influenced reactions. Using these expressions, both short-term and universal long-term asymptotes of non-stationary rate constants of multistage multisite reactions have been found, and the time range of their applicability has been discussed.

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Acknowledgements

The author is grateful to the Russian Federal Ministry of Science and High Education for financial support and to Dr. Alexander A. Kipriyanov for helpful discussions

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Appendices

Appendices

Non-stationary free resolvent of translation diffusion (non-stationary free Green function)

Contact value of free non-stationary (without reaction) translational angular Green function (non-stationary free resolvent) at diffusion motion is [40]

$$ G_{0k} \left( {w;s} \right) = G_{0k} \,\left( {R,R,\;\cos \gamma ;\;s} \right) = \,\sum\limits_{l}^{\infty } {a_{l}^{\left( k \right)} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)} \,P_{l} \;\left( {\cos \,\gamma } \right)\quad \left( {w = \sin \frac{\gamma }{2}} \right), $$
(A1)

where.

$$ a_{l}^{\left( k \right)} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right) = \frac{1}{{k_{D}^{\left( k \right)} }}\frac{{\left( {2l + 1} \right)}}{{\left( {l + 1} \right) + \sqrt {s\,\tau_{D}^{\left( k \right)} } \,\frac{{K_{{l - \frac{1}{2}}} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}}{{K_{{l + \frac{1}{2}}} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}}}}\quad \quad \left( {k_{D}^{\left( k \right)} = 4\pi R_{k} D_{k} ,\quad \tau_{D}^{\left( k \right)} = \frac{{R_{k}^{2} }}{{D_{k} }}} \right), $$
(A2)

and \(K_{\nu } \left( x \right)\) is the modified Bessel functions of the second kind (McDonald functions). Note that.

$$ \int\limits_{0}^{\pi } {G_{0k} \,\left( {R,R,\;\cos \gamma ;\;s} \right)} \sin \gamma d\gamma = 4\int\limits_{0}^{1} {G_{0k} \left( {w;s} \right)wd} w = \frac{2}{{k_{D}^{\left( k \right)} \left( {1 + \sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}}\; $$
(A3)

by virtue of orthogonality of Legendre polynomials.

Consider the asymptotic behavior of the function under the change in parameter \(\sqrt {s\,\tau_{D}^{\left( k \right)} }\). At \(\sqrt {s\,\tau_{D}^{\left( k \right)} } \to 0\) the stationary free resolvent (stationary free Green function) \((s = 0)\) \(G_{0k} \left( w \right) = G_{0k} \left( {w;0} \right) = G_{0k} \,\left( {R_{k} ,R_{k} ,\;\cos \gamma ;\;0} \right)\) is the fundamental term of the expansion. As for the next term of asymptotic expansion, it follows from the term in Eq. (A1) with \(l = 0\) which corresponds to spherically symmetric Green function, so (see Eq. A2, in view of \(K_{{ - \frac{1}{2}}} \left( s \right) = K_{\frac{1}{2}} \left( s \right)\)).

$$ - \frac{1}{{4\pi R_{k} D}}_{k} \;\sqrt {s\,\tau_{D}^{\left( k \right)} } = - \frac{\sqrt s }{{4\pi D_{k}^{3/2} }}. $$
(A4)

Coefficients in Eq. (A1) with \(l \ne 0\) are calculated from Eq. (A2) using the behavior of the modified Bessel functions of the second kind at small values of the argument [41]

$$ \frac{{K_{{l - \frac{1}{2}}} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}}{{K_{{l + \frac{1}{2}}} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}} \approx \frac{{\frac{{1 \cdot 3 \cdot 5...\left( {2l - 3} \right)}}{{\left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)^{l} }}}}{{\frac{{1 \cdot 3 \cdot 5...\left( {2l - 1} \right)}}{{\left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)^{l + 1} }}}}\sim\left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right). $$
(A5)

As is seen from Eq. (A2), they depend solely on \(\left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)^{2} = s\,\tau_{D}^{\left( k \right)}\) and higher powers of \(s\). Thus we have the expansion.

$$ G_{0} \left( {w;s} \right) \approx G_{0} \left( w \right) - \frac{\sqrt s }{{4\pi D_{k}^{3/2} }} $$
(A6)

in full agreement with universal asymptotic expansion [18].

As to resolvents \(G_{0} \left( {w;s + \alpha } \right)\), the first term of the expansion depending on the parameter \(s\) is the term linear in this parameter.

$$ \frac{{b_{k} \left( {\cos \gamma } \right)}}{{\sqrt {\alpha \tau_{D}^{\left( k \right)} } }}\;s, $$
(A7)

where.

$$ b_{k} \left( {\cos \gamma } \right) = \left. {\sum\limits_{l}^{\infty } {\frac{d}{d\,y}a_{l}^{\left( k \right)} \left( y \right)} } \right|_{{y = \alpha \tau_{D}^{\left( k \right)} }} P_{l} \;\left( {\cos \,\gamma } \right). $$
(A8)

At \(\sqrt {s\,\tau_{D}^{\left( k \right)} } \to \infty\), using the asymptotic expansion at \(x \to \infty\) [41],

$$ K_{\nu } \left( x \right)_{x \to \infty } \approx \sqrt {\frac{\pi }{\nu }} \,\exp \left( { - x} \right)\,\left\{ {1 + \frac{{4\nu^{\,2} - 1}}{8\,x} + ...} \right\}, $$
(A9)

we have.

$$ \frac{{K_{{l - \frac{1}{2}}} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}}{{K_{{l + \frac{1}{2}}} \left( {\sqrt {s\,\tau_{D}^{\left( k \right)} } } \right)}} \approx \left( {1 + \frac{{\left( {l - \frac{1}{2}} \right)^{2} - \left( {l + \frac{1}{2}} \right)^{2} }}{{2\,\sqrt {s\,\tau_{D}^{\left( k \right)} } }}} \right) = 1 - \frac{l}{{\sqrt {s\,\tau_{D}^{\left( k \right)} } }}, $$
(A10)

so Eqs. (A1), (A2), and (A10) give at \(\sqrt {s\,\tau_{D}^{\left( k \right)} } \to \infty\).

$$ \begin{aligned} & G_{0k} \,\left( {w;s} \right)_{s \to \infty } \sim\;G_{0k} \,\left( {w;s + \alpha } \right)_{s \to \infty } \sim\frac{1}{{2\pi R_{\,k} D_{k} \sqrt {s\,\tau_{D}^{\left( k \right)} } }}\,\,\sum\limits_{l = 0}^{\infty } {\frac{{\left( {2l + 1} \right)}}{2\,}} \,P_{l} \;\left( {\cos \,\gamma } \right) \\ & \; = \frac{1}{{2\,\pi R_{k} D_{k} \,\sqrt {s\,\tau_{D}^{\left( k \right)} } }}\,\delta \left( {1 - \cos \gamma } \right) = \frac{1}{{8\,\pi R_{k} \,D_{k} \,w\sqrt {s\,\tau_{D}^{\left( k \right)} } }}\,\delta \left( w \right)\quad \left( {w = \sin \frac{\gamma }{2},\quad \delta \left( {2w^{2} } \right) = \frac{\delta \left( w \right)}{{4w}}} \right). \\ \end{aligned} $$
(A11)

Behavior of conditional probability of realization of active patches orientation angles

Consider the conditional probability density \(g_{0k} \left( {\theta_{A} ,\theta_{B} ,R_{k} ;\theta_{0A} ,\theta_{0B} ,R_{k} ;s} \right)\) (following from Eq. (3.11)) of realization of polar angles \(\theta_{A}\) and \(\theta_{B}\) of orientation of active patches of reactants on contact of rigid spheres in “collision” system in the absence of rotation \(\tau_{k\,j}^{\,A} = \tau_{k\,l}^{\,B} = \infty\) of reactants on condition that \(\theta_{0A} = \theta_{0B} = 0\) (initial contact of centers of active patches). Replacing the integration variable \(\gamma\) by \(x = \cos \gamma\), we have.

$$ \begin{aligned} g_{0k} \left( {\theta_{A} ,\theta_{B} ,R_{\,k} ;0,0,R_{\,k} ;s} \right) & = \frac{\pi }{2}\sum\limits_{j,\,l = 0}^{\infty } {\left( {\int\limits_{0}^{\pi } {G_{0k} \left( {R_{\,k} ,R_{\,k} ,\,\,x;\,\,s} \right)P_{j} \left( x \right)P_{l} \left( x \right)dx} } \right)} \\ & \;\left( {2j + 1} \right)P_{j} \left( {\cos \theta_{A} } \right)\left( {2l + 1} \right)P_{l} \left( {\cos \theta_{B} } \right) \\ & \equiv 2\pi \sum\limits_{j,\,l = 0}^{\infty } {\left( {\int\limits_{ - 1}^{1} {G_{0k} \left( {R_{\,k} ,R_{\,k} ,\,\,x;\,\,s} \right)\delta \left( {x - \cos \theta_{A} } \right)\delta \left( {x - \cos \theta_{B} } \right)d\,x} } \right)} , \\ \end{aligned} $$
(B1)

where.

$$ \delta \left( {x - \cos \theta_{p} } \right) = \sum\limits_{m = 0}^{\infty } {\frac{2m + 1}{2}} P_{m} \left( x \right)P_{m} \left( {\cos \theta_{p} } \right)\quad \quad \left( {p = A,B} \right) $$
(B2)

is Dirac delta function. It is easily seen that such a probability density (at any value of the Laplace variable \(s\)) differs from zero only at \(\theta_{A} = \theta_{B}\)(see Fig. 7). So with condition (3.33), the integral of any function \(w\) involving non-overlapping functions \(\Phi_{A}^{\left( k \right)} \left( {w,\,\,\delta_{A}^{\left( k \right)} } \right)\) and \(\Phi_{B}^{\left( k \right)} \left( {w,\,\,\delta_{B}^{\left( k \right)} } \right)\) in Eq. (3.32) is equal to zero.

Calculations of functions \(\Phi_{p}^{\left( k \right)} \left( {w,\delta_{p}^{\left( k \right)} } \right)\)

As for reactants \(A_{k}\) and \(B_{k}\) functions \(\Phi_{p}^{\left( k \right)} \left( {w,\delta_{p}^{\left( k \right)} } \right)\) are of the same form for any channel, in our calculations we omit indices of reactants and channels and consider a universal function \(\Phi \left( {w,\delta } \right)\). Taking into account Eq. (3.31), definition of function \(Y\left( {w,\,\,\delta } \right)\) as functions \(S\left( {1 - 2w^{2} ,\theta ,\,\theta_{0} } \right)\) (3.20) integrated over polar angles \(\theta\) and \(\theta_{0}\) within the corresponding active patches divided by \(4f\), and Eqs. (3.26) and (3.27), we obtain.

$$ \Phi \left( {w,\delta } \right) = \frac{1}{4f}\sum\limits_{m = 0}^{\infty } {P_{m} \left( {\cos \gamma } \right)} \,P_{m} \left( {\cos \delta } \right)\,\chi_{m} ,\;\,\quad \chi_{m} = \frac{{\left( {P_{m - 1} \left( c \right) - P_{m + 1} \left( c \right)} \right)^{2} }}{2m + 1}\quad \left( {c = 1 - 2f} \right). $$
(C1)

Using the Legendre addition theorem, we have Eq. (3.34a) where, in view of Eqs. (3.28) and (3.35),

$$ y = \cos \delta \cos \gamma + \sin \delta \sin \gamma \cos \varphi \equiv \left( {1 - 2w^{2} } \right)\cos \delta + 2w\sqrt {1 - w^{2} } \sin \delta $$
(C2)

and the function given in Eq. (3.34b) is the result of summation.

$$ W\left( y \right) = \frac{1}{4f}\sum\limits_{m = 0}^{\infty } {P_{m} \left( y \right)} \frac{{\left( {P_{m - 1} \left( c \right) - P_{m + 1} \left( c \right)} \right)^{2} }}{2m + 1}\,. $$
(C3)

To make summation in Eq. (C3), let us express the square of difference of Legendre polynomials in terms of the sum of their squares using the expression for the product \(P_{m - 1} \left( c \right)P_{m + 1} \left( c \right)\) obtained by raising the following recurrent relation to the square.

$$ m\,P_{m - 1} \left( c \right) + \left( {m + 1} \right)P_{m + 1} \left( c \right) = \left( {2m + 1} \right)c\,P_{m} \left( c \right). $$
(C4)

Expressing the squares of Legendre polynomials in terms of Legendre polynomials by the formula.

$$ P_{m}^{2} \left( c \right) = \frac{1}{\pi }\int\limits_{0}^{\pi } {P_{m} \left( {c^{2} + \left( {1 - c^{2} } \right)\cos \varphi } \right)d\varphi } , $$
(C5)

following from Legendre addition theorem, and using again the recurrent relation C6 (where \(c\) is replaced by \(c^{2} + \left( {1 - c^{2} } \right)\cos \varphi )\), we derive.

$$ \left( {P_{m - 1} \left( c \right) - P_{m + 1} \left( c \right)} \right)^{2} = \left\{ {\begin{array}{*{20}c} {\frac{{1 - c^{2} }}{\pi }\int\limits_{0}^{\pi } {\frac{{\left( {2m + 1} \right)^{2} }}{{m\left( {m + 1} \right)}}P_{m} \left( {c^{2} + \left( {1 - c^{2} } \right)\cos \varphi } \right)\cos \varphi d\varphi \quad {\text{at }}\quad m \ne 0} } \\ {\left( {1 - c} \right)^{2} = 4f^{2} \quad \quad \quad \quad {\text{at }}\;\quad m = 0\quad \quad \quad } \\ \end{array} } \right.. $$
(C6)

Using Eq. (B6) from Eq. (C3) after changing of integration variable, one has.

$$ W\left( y \right) = f + \frac{2}{\pi }\left( {1 - f} \right)\int\limits_{0}^{{\frac{\pi }{2}}} {\cos 2\varphi \,d\varphi \;\left\{ {\sum\limits_{m = 1}^{\infty } {\frac{2m + 1}{{m\left( {m + 1} \right)}}\,P_{m} \left( y \right)\,P_{m} \left( {z^{2} + \left( {1 - z^{2} } \right)\cos 2\varphi } \right)} } \right\}} . $$
(C7)

Now we employ the tabulated sum.

$$ \sum\limits_{m = 1}^{\infty } {\frac{2m + 1}{{m\left( {m + 1} \right)}}\,P_{m} \left( x \right)\,P_{m} \left( z \right)} = 2\ln 2 - 1 - \ln \left[ {\left( {1 - x} \right)\,\left( {1 + z} \right)} \right]\quad - 1 < x \le z < 1. $$
(C8)

Let us divide the interval of integration with respect to \(\varphi\) in Eq. C7 into the interval \(\varphi^{0} \le \varphi \le \frac{\pi }{2}\) where.

$$ x = c^{2} + \left( {1 - c^{2} } \right)\cos 2\varphi = 1 - 2\left( {1 - c^{2} } \right)\sin^{2} \varphi \le z = y \le 1\;\quad {\text{or }}\quad \sin \varphi \ge \sin \varphi^{0} \; $$
(C9)

and the interval \(0 \le \varphi \le \varphi^{0}\) where.

$$ x = y \le c^{2} + \left( {1 - c^{2} } \right)\cos 2\varphi = z \le 1\quad {\text{or}}\quad \sin \varphi \le \sin \varphi^{0} . $$
(C10)

In Eqs. (C9) and (C10).

$$ \sin \varphi^{0} = \frac{1}{a}\sqrt {\frac{1 - y}{2}} \;\quad \left( {2c^{2} - 1 = 1 - 2a^{2} \le y \le 1,\quad c^{2} = 1 - a^{2} } \right). $$
(C11)

Integrals in (C7), in view of Eq. (C8), at \(2c^{2} - 1 = 1 - 2a^{2} \le y \le 1\) are as follows.

$$ W\left( y \right) = 1 - \frac{4}{\pi }\left( {1 - f} \right)\int\limits_{0}^{{\varphi^{0} }} {\frac{{\cos^{2} \varphi }}{{1 - \left( {1 - c^{2} } \right)\sin^{2} \varphi }}\,d\varphi } = 1 - \frac{4}{\pi }\left( {1 - f} \right)\int\limits_{0}^{{\tan \varphi^{0} }} {\frac{d\,t}{{\left( {1 + t^{2} } \right)\left( {1 + c^{2} t^{2} } \right)}}} . $$
(C12)

At \(y \le 2c^{2} - 1 = 1 - 2a^{2}\) in the integral in Eq. (C12) one should put \(\varphi^{0} = \frac{\pi }{2}\). As a result of integration, we get Eq. (3.34b) where \(b = \left| c \right|\).

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Doktorov, A.B. Long-term and short-term asymptotes of reaction rate constants in the presence of reactivity anisotropy. J Math Chem (2021). https://doi.org/10.1007/s10910-021-01295-7

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Keywords

  • Diffusion-influenced reactions
  • Multistage multisite reactions
  • Reaction rate constants
  • Rreactivity anisotropy
  • Sshort-term and long-term asymptotes
  • Mmixing principle